Find the slope of a line perpendicular to the line whose equation is \( x+4 y=28 \). Fully simplify your answer.

To find the slope of a line perpendicular to the given line \( x + 4y = 28 \), we first need to rewrite the equation in slope-intercept form \( y = mx + b \), where \( m \) represents the slope. Starting with \( x + 4y = 28 \), we can isolate \( y \): 1. Subtract \( x \) from both sides: \[ 4y = -x + 28 \] 2. Divide every term by 4 to solve for \( y \): \[ y = -\frac{1}{4}x + 7 \] The slope \( m \) of this line is \( -\frac{1}{4} \). For a line to be perpendicular, its slope must be the negative reciprocal of the original slope. This means we flip the fraction and change the sign: The negative reciprocal of \( -\frac{1}{4} \) is \( 4 \). Thus, the slope of a line perpendicular to the line given by \( x + 4y = 28 \) is \( 4 \).